Holomorphic discrete models of semisimple Lie groups and their symplectic constructions
type: Item
instance of: scholarly article
title: Holomorphic discrete models of semisimple Lie groups and their symplectic constructions
zbMATH Open document ID: 0960.22015
author: Meng-Kiat Chuah
published in: Journal of Functional Analysis
publication date: 2001-05-10T00:00:00Z
review text: Let \(G\) be a connected real semisimple Lie group. An irreducible unitary representation of \(G\) is in the discrete series if some non-zero matrix coefficient function is in \(L^2(G)\). The author defines the holomorphic discrete model as a unitary \(G\)-representation consisting of all holomorphic discrete series with multiplicity one. He constructs such an object when \(G\) has a compact Cartan subgroup (e.g. \(G=SL(2,\mathbb{R})\), \(SU(p,q)\) etc.) by using symplectic techniques. In this case, if \(B\) is a Borel subgroup of the complexification \(G^\mathbb{C}\) of \(G\) which contains a compact Cartan subgroup of \(G\), the set \(G\cdot B\) is open in \(G^\mathbb{C}\) and the \(G\)-orbit of \(eP\) in \(G^C/P\), where \(P\supset B\) is a parabolic subgroup of \(G^\mathbb{C}\), is an open \(G\)-orbit. The construction of the holomorphic discrete model is achieved by studying line bundles over the inverse images of these distinguished \(G\)-orbits in flag manifolds \(G^\mathbb{C}/P\) via the fibration \(G^\mathbb{C}/(P,P) \to G^\mathbb{C}/P\), and by using the machinery of geometric quantization. The paper has many nice ideas and the author has taken pains in explaining these ideas and the methodology of the proofs.
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 1507744
zbMATH Keywords: real semisimple Lie group
zbMATH Keywords: irreducible unitary representation
zbMATH Keywords: discrete series
zbMATH Keywords: holomorphic discrete model
zbMATH Keywords: line bundles
zbMATH Keywords: flag manifolds
reviewed by: Hassan Azad
MaRDI profile type: Publication
OpenAlex ID: W2040589490
cites work: Kählerian Coset Spaces of Semisimple Lie Groups
cites work: 𝐾-invariant Kaehler structures on 𝐾_{𝐂}/𝐍 and the associated line bundles
cites work: Kaehler structures on 𝐾_{𝐂}/(𝐏,𝐏)
cites work: Q4278097
cites work: Models of representations of classical groups and their hidden symmetries
cites work: Geometric quantization and multiplicities of group representations
cites work: Q3696183
cites work: Representations of Semisimple Lie Groups VI: Integrable and Square-Integrable Representations
cites work: Discrete series for semisimple Lie groups. II: Explicit determination of the characters
cites work: Q3897665
cites work: Q3331943
cites work: Q3741813
cites work: Quantization and unitary representations
cites work: Reduction of symplectic manifolds with symmetry
cites work: \(L^2\)-cohomology and the discrete series
cites work: Symplectic reduction and Riemann-Roch formulas for multiplicities
DOI: 10.1006/JFAN.2000.3586
description: scientific article; zbMATH DE number 1507744